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Algebra tiles can be used to model polynomials. These 1 -by- 1 square tiles have an area of 1 square unit. These 1 -by- x rectangular tiles have an area of x square units. These x -by- x rectangular tiles have an area of x 2 square units. +– +–+– 1–1x–xx 2x 2 –x 2 MODELING ADDITION OF POLYNOMIALS

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You can use algebra tiles to add the polynomials x 2 + 4x + 2 and 2 x 2 – 3x – 1. ++ – – MODELING ADDITION OF POLYNOMIALS +++ + + ++–– 1Form the polynomials x 2 + 4x + 2 and 2 x 2 – 3x – 1 with algebra tiles. x 2 + 4x + 2 2 x 2 – 3x – 1

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MODELING ADDITION OF POLYNOMIALS You can use algebra tiles to add the polynomials x 2 + 4x + 2 and 2 x 2 – 3x – 1. +++++ + +– –++–– x 2 + 4x + 2 2x 2 – 3x – 1 2To add the polynomials, combine like terms. Group the x 2 -tiles, the x -tiles, and the 1 -tiles. + ++ + + ++++ ––– + +– =

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MODELING ADDITION OF POLYNOMIALS You can use algebra tiles to add the polynomials x 2 + 4x + 2 and 2 x 2 – 3x – 1. +++++ + +– –++–– x 2 + 4x + 2 2x 2 – 3x – 1 2To add the polynomials, combine like terms. Group the x 2 -tiles, the x -tiles, and the 1 -tiles. + ++ + + ++++ ––– + +– = 3 Find and remove the zero pairs. The sum is 3x 2 + x + 1.

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An expression which is the sum of terms of the form a x k where k is a nonnegative integer is a polynomial. Polynomials are usually written in standard form. Adding and Subtracting Polynomials Standard form means that the terms of the polynomial are placed in descending order, from largest degree to smallest degree. The degree of each term of a polynomial is the exponent of the variable. Polynomial in standard form: 2 x 3 + 5x 2 – 4 x + 7 DegreeConstant termLeading coefficient The degree of a polynomial is the largest degree of its terms. When a polynomial is written in standard form, the coefficient of the first term is the leading coefficient.

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A polynomial with only one term is called a monomial. A polynomial with two terms is called a binomial. A polynomial with three terms is called a trinomial. Identify the following polynomials: Classifying Polynomials PolynomialDegree Classified by degree Classified by number of terms 6 –2 x 3x + 1 –x 2 + 2 x – 5 4x 3 – 8x 2 x 4 – 7x 3 – 5x + 1 0 1 1 4 2 3 constant linear quartic quadratic cubic monomial binomial polynomial trinomial binomial

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Find the sum. Write the answer in standard format. (5x 3 – x + 2 x 2 + 7) + (3x 2 + 7 – 4 x) + (4x 2 – 8 – x 3 ) Adding Polynomials SOLUTION Vertical format: Write each expression in standard form. Align like terms. 5x 3 + 2 x 2 – x + 7 3x 2 – 4 x + 7 – x 3 + 4x 2 – 8 + 4x 3 + 9x 2 – 5x + 6

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Find the sum. Write the answer in standard format. (2 x 2 + x – 5) + (x + x 2 + 6) Adding Polynomials SOLUTION Horizontal format: Add like terms. (2 x 2 + x – 5) + (x + x 2 + 6) =(2 x 2 + x 2 ) + (x + x) + (–5 + 6) =3x 2 + 2 x + 1

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Find the difference. (–2 x 3 + 5x 2 – x + 8) – (–2 x 2 + 3x – 4) Subtracting Polynomials SOLUTION Use a vertical format. To subtract, you add the opposite. This means you multiply each term in the subtracted polynomial by –1 and add. –2 x 3 + 5x 2 – x + 8 –2 x 3 + 3x – 4– Add the opposite No change –2 x 3 + 5x 2 – x + 8 2 x 3 – 3x + 4 +

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Find the difference. (–2 x 3 + 5x 2 – x + 8) – (–2 x 2 + 3x – 4) Subtracting Polynomials SOLUTION Use a vertical format. To subtract, you add the opposite. This means you multiply each term in the subtracted polynomial by –1 and add. –2 x 3 + 5x 2 – x + 8 –2 x 3 + 3x – 4– 5x 2 – 4x + 12 –2 x 3 + 5x 2 – x + 8 2 x 3 – 3x + 4 +

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Find the difference. (3x 2 – 5x + 3) – (2 x 2 – x – 4) Subtracting Polynomials SOLUTION Use a horizontal format. (3x 2 – 5x + 3) – (2 x 2 – x – 4)= (3x 2 – 5x + 3) + (–1)(2 x 2 – x – 4) = x 2 – 4x + 7 = (3x 2 – 5x + 3) – 2 x 2 + x + 4 = (3x 2 – 2 x 2 ) + (– 5x + x) + (3 + 4)

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Total Area = (10x)(14x – 2) (square inches) Area of photo = You are enlarging a 5 -inch by 7 -inch photo by a scale factor of x and mounting it on a mat. You want the mat to be twice as wide as the enlarged photo and 2 inches less than twice as high as the enlarged photo. Using Polynomials in Real Life Write a model for the area of the mat around the photograph as a function of the scale factor. Verbal Model Labels Area of mat = Area of photo Area of mat = A (5x)(7x) (square inches) Total Area – Use a verbal model. 5x5x 7x7x 14x – 2 10x SOLUTION …

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(10x)(14x – 2) – (5x)(7x) You are enlarging a 5 -inch by 7 -inch photo by a scale factor of x and mounting it on a mat. You want the mat to be twice as wide as the enlarged photo and 2 inches less than twice as high as the enlarged photo. Using Polynomials in Real Life Write a model for the area of the mat around the photograph as a function of the scale factor. A = = 140x 2 – 20x – 35x 2 SOLUTION = 105x 2 – 20x A model for the area of the mat around the photograph as a function of the scale factor x is A = 105x 2 – 20x. Algebraic Model … 5x5x 7x7x 14x – 2 10x

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